User:SBLCebry/sandbox

Tensile fractures (NOTE: Includes some info from original post)
Rocks contain many pre-existing cracks where development of tensile fracture, or Mode I fracture, may be examined.

The first form is in axial stretching. In this case a remote tensile stress, σn, is applied, allowing microcracks to open slightly throughout the tensile region. As these cracks open up, the stresses at the crack tips intensify, eventually exceeding the rock strength and allowing the fracture to propagate. This can occur at times of rapid overburden erosion. Folding also can provide tension, such as along the top of an anticlinal fold axis. In this scenario the tensile forces associated with the stretching of the upper half of the layers during folding can induce tensile fractures parallel to the fold axis.

Another, similar tensile fracture mechanism is hydraulic fracturing. In a natural environment, this occurs when rapid sediment compaction, thermal fluid expansion, or fluid injection causes the pore fluid pressure, σp, to exceed the pressure of the least principal normal stress, σn. When this occurs, a tensile fracture opens perpendicular to the plane of least stress.[4]

Tensile fracturing may also be induced by applied compressive loads, σn, along an axis such as in a Brazilian disk test. This applied compression force results in longitudinal splitting. In this situation, tiny tensile fractures form parallel to the loading axis while the load also forces any other microfractures closed. To picture this, imagine an envelope, with loading from the top. A load is applied on the top edge, the sides of the envelope open outward, even though nothing was pulling on them. Rapid deposition and compaction can sometimes induce these fractures.

Tensile fractures are almost always referred to as joints, which are fractures where no appreciable slip or shear is observed.

To fully understand the effects of applied tensile stress around a crack in a brittle material such a rock, fracture mechanics can be used. The concept of fracture mechanics was initially developed by A. A. Griffith during World War I. Griffith looked at the energy required to create new surfaces by breaking material bonds versus the elastic strain energy of the stretched bonds released. By analyzing a rod under uniform tension Griffith determined an expression for the critical stress at which a favorably orientated crack will grow. The critical stress at fracture is given by,

$$\sigma_f=({2E\gamma \over \pi a})^{1/2}$$

where γ = surface energy associated with broken bonds, E = Young's modulus, and a = half crack length. It should be noted that fracture mechanics has generalized to that γ represents energy dissipated in fracture not just the energy associated with creation of new surfaces

Linear Elastic Fracture Mechanics
Linear elastic fracture mechanics builds off the energy balance approach taken by Griffith but provides a more generalized approach for many crack problems. LEFM investigates the stress field near the crack tip and bases fracture criteria on stress field parameters. One important contribution of LEFM is the stress intensity factor, K, which is used to predict the stress at the crack tip. The stress field is given by

$$\sigma_{ij}(r, \theta)={K\over(2\pi r)^{1/2}}f_{ij}(\theta)$$

where $$ K$$ is the stress intensity factor for Mode I, II, or III cracking and $$   f_{ij}$$is a dimensionless quantity that varies with applied load and sample geometry. As the stress field gets close to the crack tip, i.e. $$ r\rightarrow0$$, $$   f_{ij}$$becomes a fixed function of $$ \theta$$. With knowledge of the geometry of the crack and applied far field stresses, it is possible to predict the crack tip stresses, displacement, and growth. Energy release rate is defined to relate K to the Griffith energy balance as previously defined. In both LEFM and energy balance approaches, the crack is assumed to be cohesionless behind the crack tip. This provides a problem for geological applications such a a fault, where friction exists all over a fault. Overcoming friction absorbs some of the energy that would otherwise go to crack growth. This means that for Modes II and III crack growth, LEFM and energy balances represent local stress fractures rather than global criteria.

Crack Formation and Propagation
Cracks in rock do not form smooth path like a crack in a car windshield or a highly ductile crack like a ripped plastic grocery bag. Rocks are a polycrystalline material so cracks grow through the coalescing of complex microcracks that occur in front of the crack tip. This area of microcracks is called the brittle process zone. Consider a simplified 2D shear crack as shown in the image on the right. The shear crack, shown in blue, propagates when tensile cracks, shown in red, grow perpendicular to the direction of the least principal stresses. The tensile cracks propagate a short distance then become stable, allowing the shear crack to propagate. This type of crack propagation should only be considered an example. Fracture in rock is a 3D process with cracks growing in all directions. It is also important to note that once the crack grows, the microcracks in the brittle process zone are left behind leaving a weakened section of rock. This weakened section is more susceptible to changes in pore pressure and dilatation or compaction. Note that this description of formation and propagation considers temperatures and pressures near the Earth's surface. Rocks deep within the earth are subject to very high temperatures and pressures. This causes them to behave in the semi-brittle and plastic regimes which result in significantly different fracture mechanisms. In the plastic regime cracks acts like a plastic bag being torn. In this case stress at crack tips goes to two mechanisms, one which will drive propagation of the crack and the other which will blunt the crack tip. In the brittle-ductile transition zone, material will exhibit both brittle and plastic traits with the gradual onset of plasticity in the polycrystalline rock. The main form of deformation is called cataclastic flow, which will cause fractures to fail and propagate due to a mixture of brittle-frictional and plastic deformations.

Faults and shear fractures (add info and figures to existing section)
Add under frictional sliding: It should be noted that the shear force required to slip fault is less than force required to fracture and create new faults as shown by the Mohr-Coulumb diagram. Since the earth is full of existing cracks and this means for any applied stress, many of these cracks are more likely to slip and redistribute stress than a new crack is to initiate. The Mohr's Diagram shown, provides a visual example. For a given stress state in the earth, if an existing fault or crack exists orientated anywhere from −α/4 to +α/4, this fault will slip before the strength of the rock is reached and a new fault is formed. While the applied stresses may be high enough to form a new fault, existing fracture planes will slip before fracture occurs.